Euler–Mascheroni constant

The area of the blue region converges to the Euler–Mascheroni constant.

The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter Template:Lang (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
).

History

The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Eneström Index 43). Euler used the notations C and O for the constant. In 1790, Italian mathematician Lorenzo Mascheroni used the notations A and a for the constant. The notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time perhaps because of the constant's connection to the gamma function.[3] For example, the German mathematician Carl Anton Bretschneider used the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
in 1835[4] and Augustus De Morgan used it in a textbook published in parts from 1836 to 1842.[5]

Appearances

The Euler–Mascheroni constant appears, among other places, in the following ('*' means that this entry contains an explicit equation):

In expressions revealing the key properties of an exoplanet atmosphere (temperature, pressure, and composition) embedded in its absorption spectrum, which are at the basis of a new method to determine the mass of exoplanets, MassSpec.[6]

Properties

The number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
has not been proved algebraic or transcendental. In fact, it is not even known whether Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is irrational. Continued fraction analysis reveals that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is rational, its denominator must be greater than 10242080.[7] The ubiquity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
revealed by the large number of equations below makes the irrationality of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
a major open question in mathematics. Also see Sondow (2003a).

Relation to gamma function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is related to the digamma function Ψ, and hence the derivative of the gamma function Γ, when both functions are evaluated at 1. Thus:

Integrals

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
equals the value of a number of definite integrals:

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): H_{{x}}
is the fractional Harmonic number.

One can express Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
using a special case of Hadjicostas's formula as a double integral (Sondow 2003a, 2005) with equivalent series:

It shows that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \ln \left({\frac {4}{\pi }}\right)
may be thought of as an "alternating Euler constant".

The two constants are also related by the pair of series (see Sondow 2005 #2)

Series expansions

Euler showed that the following infinite series approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
:

The series for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is equivalent to series Nielsen found in 1897:

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \log _{2}
is the logarithm of base 2 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \lfloor \,\rfloor
is the floor function.

has a deep connection with Euler's constant and was studied by James Gregory in connection with numerical integration. The coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): C_{n}
are called Gregory coefficients; the first six were given in a letter to John Collins in 1670. From the equations

, which can be used recursively to get these coefficients for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): n\geq 1
, we get the table

eγ

The constant eγ is important in number theory. Some authors denote this quantity simply as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma ^{\prime }
. eγ equals the following limit, where pn is the nth prime number:

This infinite product, first discovered by Ser in 1926, was rediscovered by Sondow (2003) using hypergeometric functions.

Continued fraction

The continued fraction expansion of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is of the form [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, ...] Template:OEIS2C, of which there is no apparent pattern. The continued fraction has at least 470,000 terms,[7] and it has infinitely many terms if and only ifFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
is irrational.

for 0 < α < 1, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma
as the special case α = 1.[10] This can be further generalized to

Published digits

Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Mascheroni attempted to calculate the constant to 32 decimal places, but made errors in the 20th–22nd decimal places; starting from the 20th digit, he calculated ...1811209008239 when the correct value is ...0651209008240.

Published Decimal Expansions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): \gamma